Ch-4pic.pdf

 

 f  c =  f  c 1 −  / 

0.15 0.003 − εo

 

(εc − εo )

f c′

Stress

 εc  εc  2   f  c = f  c 2 −        εo  εo     / 

Strain

ε0

0.003

Figure 4.1: Theoretical Stress-Strain Plot of Concrete

f s = f y Es Stress f s = εsEs

εy

Strain

Figure 4.2: Theoretical Theoretical Stress-Strain Plot of Steel Reinforcement

44

b

ε ds

αf  c

EC

f s

′

εs

c

γ c

Cc Cs

ds

dCFRP

εs ε

f s

CFRP

Section

Ts TCFRP

f CFRP Strain

Stress

Force

Figure 4.3: Strain, Stress, and Force Diagrams of CFRP Strengthened Concrete Beam

f c′ f c Stress dεc

εEC

Strain

0.003

Figure 4.4: Integration of Concrete Stress-Strain Curve

45

εc E

R

Neutral Axis

c

ϕ

M M ds

ε

S

Tension Steel Section

Strain Distribution

Figure 4.5: Rotation of Flexural Member

700 600   s 500   p    i    k     n 400    i  ,    t   n   e 300   m   o    M200

Control Two-Layer Three-Layer

100

Four-Layer

0 0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Deflection, in.

Figure 4.6: Theoretical Moment-Deflection Plots of 0° /90° CFRP Strengthened Beams

46

700 600   s 500   p    i    k     n 400    i  ,    t   n   e 300   m   o    M200

Control Two-Layer Three-Layer Four-Layer

100 0 0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Deflection, in.

Figure 4.7: Theoretical Moment-Deflection Plots of ±45° CFRP Strengthened Beams

700 600

  s 500   p    i    k     n 400    i  ,    t   n   e 300   m   o    M200

Two-Layer Three-Layer Four-Layer

100 0 0

0.002

0.004

0.006

0.008

0.01

CFRP Strain, in/in

Figure 4.8: Theoretical Moment-CFRP Strain Plots of 0 ° /90° CFRP Strengthened Beams

47

700 600

  s 500   p    i    k   - 400   n    i  ,    t   n   e 300   m   o    M200

Two-Layer Three-Layer Four-Layer

100 0 0

0.002

0.004

0.006

0.008

0.01

CFRP Strain, in/in

Figure 4.9: Theoretical Moment-CFRP Strain Plots of ±45° CFRP Strengthened Beams

48

Steel Reinforcement

FRP Reinforcement

Moment

Moment

Elastic Energy Released at Failure

Elastic Energy Released at Failure Deflection

Deflection

Figure 4.10: Comparison of Elastic Energy Released at Failure

EINEL, Inelastic Energy Consumed Prior to Failure MU M2 Moment

EEL, Elastic Energy Released at Failure

S2

* ETOT = EEL + EINEL

M1

S *S= S1

M1 S1 + (M2 - M1) S2 M2

* S, S1, S2: Slopes of Lines Deflection

Figure 4.11: Approximation Process for Energy Method

49